{"id":"n-227b620650ca6bf3","revision":0,"first_index":0,"fingerprint":"f6-7b312ef315523524","name":"shift_x_plus_one of Triangle read by rows: T(n,k) = 2^k * (n!/k!)*binomial(n-1,k-1).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["2"],["8","4"],["44","40","8"],["304","384","144","16"],["2512","3968","2112","448","32"],["24064","44960","30400","9280","1280","64"],["261536","558528","452160","177280","35520","3456","128"],["3173888","7571200","7074816","3351040","869120","123648","8960","256"],["42483968","111351808","117168128","64540672","20357120","3770368","401408","22528","512"],["621159424","1766877696","2056679424","1283874816","473647104","107218944","14923776","1234944","55296","1024"],["9841950208","30091965440","38235571200","26549944320","11155845120","2966777856","506634240","54988800","3640320","133120","2048"],["167879268352","547534524416","751602903040","572443576320","268738990080","81720274944","16451162112","2197155840","191262720","10362880","315392","4096"],["3065723549696","10599516733440","15589649129472","12882146140160","6659529338880","2270590451712","524295045120","82769854464","8886620160","634388480","28655616","737280","8192"],["59651093528576","217502529826816","340459156217856","302578192277504","170307831316480","64147228434432","16655060877312","3023389704192","384528089088","33919508480","2021933056","77316096","1703936","16384"],["1231571119812608","4715266036375552","7811463604625408","7413713483251712","4502356984373248","1851785053454336","532524584976384","108947699367936","15965257211904","1671182729216","123281686528","6229164032","204259328","3899392","32768"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["2","12","92","848","9072","110048","1488608","22172928","360096512","6325672448","119407452672","2408769800192","51683770652672","1174745030750208","28185394725220352"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2"]],"recurrence":{"denominator":{"coeffs":[["2"]]},"terms":[{"coeff":{"coeffs":[["8","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["6","4"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","1"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184467,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-43a96831412e78bb"]},{"id":184468,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-43a96831412e78bb"]},{"id":184469,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-43a96831412e78bb"]}],"region_exact":{},"incoming_edges":[{"id":"e-43a96831412e78bb","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-227b620650ca6bf3","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"9e9d390727c9117c627a0f36f38c363f2a6af21191dd1710f4fc8abb987a1bb3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-cbd7c2e66114e926","role":"sequence","name":"Triangle read by rows: T(n,k) = 2^k * (n!/k!)*binomial(n-1,k-1).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:         2;\n  1:         8,          4;\n  2:        44,         40,          8;\n  3:       304,        384,        144,         16;\n  4:      2512,       3968,       2112,        448,        32;\n  5:     24064,      44960,      30400,       9280,      1280,        64;\n  6:    261536,     558528,     452160,     177280,     35520,      3456,      128;\n  7:   3173888,    7571200,    7074816,    3351040,    869120,    123648,     8960,     256;\n  8:  42483968,  111351808,  117168128,   64540672,  20357120,   3770368,   401408,   22528,   512;\n  9: 621159424, 1766877696, 2056679424, 1283874816, 473647104, 107218944, 14923776, 1234944, 55296, 1024;\n","recurrence_sentence":"The row polynomials satisfy 2*P(n,x) = (8+4*x)*P(n-1,x) + (6+4*x)*d/dx P(n-1,x) + (1+x)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 2.","mathematica":"P[0, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((8 + 4*x)*P[n - 1, x] + (6 + 4*x)*D[P[n - 1, x], x] + (1 + x)*D[P[n - 1, x], {x, 2}])/(2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2; for(n = 1, N, my(i = n + 1); v[i] = ((8 + 4*x)*v[i - 1] + (6 + 4*x)*deriv(v[i - 1], x) + (1 + x)*deriv(deriv(v[i - 1], x), x))/(2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-cbd7c2e66114e926","steps":["x \u21a6 x+1"],"edges":["e-43a96831412e78bb"],"text":"A086915 \u2192 x \u21a6 x+1"},"headline":"A086915 \u2192 x \u21a6 x+1","chain":null}